Question

The square of the distance from the origin to the point of intersection of the pair of lines \( ax^2 + 2hxy - ay^2 + 2gx + 2fy + c = 0 \) is:

• A. \( \frac{f^2 + g^2}{a^2 + h^2} \)
• B. \( \frac{f^2 + g^2}{a^2 - h^2} \)
• C. \( \frac{f^2 + g^2}{h^2 - a^2} \)
• D. \( \frac{f^2 - g^2}{h^2 - a^2} \)

Hint:

For problems involving the intersection of lines and the distance from the origin, use the standard formula derived from the geometry of the conic section.

Answer 1

The Correct Option is A

Step 1: Use the formula for the square of the distance from the origin to the intersection of two lines.
The equation of the pair of lines can be written in the general form: \[ ax^2 + 2hxy - ay^2 + 2gx + 2fy + c = 0 \] The square of the distance from the origin to the point of intersection of these lines is given by the formula: \[ \frac{f^2 + g^2}{a^2 + h^2} \]
Step 2: Understand the formula derivation.
This formula comes from the general properties of the intersection point of two lines and the geometry of the quadratic equation representing the lines.